Dynamic Resources
Using Dynamic Representations
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All highlighted points and arrows are interactive (top right).
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Arrays
Multiplication and Division
The array model (above) uses double sided counters. Yellow positive and red negative. See integers for more details.
This array model partitions when the divisor or the quotient are greater than 10. This model can be used to show the connection between the array, multiplication and division. It represents the structure behind the bus stop!
Multiplication as Scaling
Area and Bar Models
Multiple Representations
Multiple representations allow students to gain a deeper understanding of the mathematical structure.
Comparing multiple methods allows students make connections and use the most appropriate method with fluency.
Comparing different representations uncovers the structure of the mathematics and allows students to make generalisations.
Fractions, Decimals and Percentages
Fractions, decimals and percentages are represented by a 100 grid, decimal number line, percentage number line and bar model. Instruction video link
FDP.mp4Percentage increase and decrease
Bar model, double number line and ratio table. Instruction video link.
Factors
Common Factors
Finding common factors using area models. In this model the height is the common factor. Change the slider to change the height. They have common factors when both bases are integers.
Area Models for Multiplication
Multiply Decimals
Multiply decimals between 0 and 1 using the area model.
Click and drag the coloured Dienes blocks over the shaded area to count the number of 0.1's, 0.01's, 0.001's and 0.0001's.
Multiply whole numbers and decimals
Multiply two decimals up to 1 decimal place using the area model. Click and drag the coloured Dienes blocks over the shaded area to count the number of 1's, 0.1's and 0.01's.
Area and Grid Models
Compare with algebra tiles
Division with Dienes and Area Model
Comparing the part whole model, bus stop method and the area model.
Quotative division (grouping)
Groups are represented by columns.
Quotative division (grouping)
Partitive division (sharing)
These representations for base 10 use the same colours as algebra tiles for ones, x and x squared including red tiles for negatives.
Quadratic Expressions with Algebra Tiles
(x+a)(x+b)
a(x+c)=d
(ax + b)(cx + d)
Multiplication of two linear factors is modelled as an area using dynamic algebra tiles.
x is variable in all models. See the model below to link the area model with the graphical representation.
Negatives are represented by red tiles and make zero pairs. In the three models above and left, you can vary the size of a, b, c, d and x.
Difference of Two Squares
Change a and b to change the size of the squares. The a+b and a-b labels are draggable to allow students to place them by the correct side.
Completing the square model to the left allows students to see the structure of the general form of x^2+bx + c = (x+b/2)^2 - (b/2)^2 + c.
(b/2)^2 and -(b/2)^2 make a zero pair.
Base Blocks (3D Algebra Tiles)
Base Blocks from Base -10 to 10
Base Blocks with Negative Indices
Base blocks allow students to make the link between Dienes blocks (base 10) and algebra tiles (base x) by exposing the mathematical structure of the place value of the base system.
Base x Interactive Blocks
Base 2 Interactive Blocks
Base 3 Interactive Blocks
Dynamic Algebra Tiles
Dynamic Variable x
Linear Expressions with Dynamic Algebra Tiles
variable variable with expressions cropped.mp4Compare expressions with algebra tile representations using the dynamic variable sliders to change x.
This representation models the x and 1 algebra tiles. The green tile x is a variable and can be changed with the slider under the x axis.
Equations and Inequalities
Sequences and Graphs
Sequences using Cuisenaire Rods
Sequences using Dynamic Algebra Tiles
sequences with table cropped.mp4These dynamic Cuisenaire rods show the first 5 terms and the nth of linear sequences of the form mn+c. Connections can be make to graphical representations and y=mx+c.
y = mx + c with Dynamic Algebra Tile
Change m, x and c using the sliders.
x + 3 and 3x
Which is bigger?
Quadratic Graphs with Dynamic Algebra Tiles
Integers
Adding Integers as Vectors
Counters and Zero Pairs
Counters, Vectors and Bar Model
Subtracting Integers - Additive Inverse
Subtracting Integers - Additive Inverse
Multiplying Integers using Vectors
multiply integers on a number line.mp4
Trigonometry
The Unit Circle
Trig with unit circle cropped.mp4Area
Area of a Circle
See Perimeter, Area and Volume for further examples
Number Lines
Variable number lines
Arithmetic and Algebra in Early Mathematics Education Adapted from D.W. Carraher et al. (2006).
Double Number Line and Ratio Tables
Speed 1
Speed 2
Double Number Line with multipliers
Statistics
Counters, Bar Chart, Bar Model and Pie Chart
